Given a specific group, up to isomorphism, is there a way to determine a topological space, up to homeomorphism, with said group as the nth homology?
In other words, is there an established algorithm to work backwards from a specific group (like $\mathbb{Z}_2$) and end up with some topological space?
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.